Quantum Mechanics for Mathematicians

wheref 1 andf 2 are complex linear combinations of the powers of the anticom-
muting variablesθj. For the details of the construction of this inner product,
see chapter 7.2 of [90] or chapters 7.5 and 7.6 of [109]. We will denote this
state space asFd+and refer to it as the fermionic Fock space. Since it is finite
dimensional, there is no need for a completion as in the bosonic case.
The quantization using fermionic annihilation and creation operators given
here provides an explicit realization of a representation of the Clifford algebra
Cliff(2d,R) on the complex vector spaceFd+. The generators of the Clifford
algebra are identified as operators onFd+by

γ 2 j− 1 =

√

2Γ+(ξ 2 j− 1 ) =

√

2Γ+

(

1

√

2

(θj+θj)

)

=aFj+a†Fj

γ 2 j=

√

2Γ+(ξ 2 j) =

√

2Γ+

(

i

√

2

(θj−θj)

)

=i(a†Fj−aFj)

Quantization of the pseudo-classical fermionic oscillator Hamiltonianhof
section 30.3.2 gives

Γ+(h) = Γ+



−i

2

∑d

j=1

(θjθj−θjθj)



=−i

2

∑d

j=1

(a†FjaFj−aFja†Fj) =−iH

(31.4)
whereHis the Hamiltonian operator for the fermionic oscillator used in chapter
27.
Taking quadratic combinations of the operatorsγjprovides a representation
of the Lie algebraso(2d) =spin(2d). This representation exponentiates to a
representation up to sign of the groupSO(2d), and a true representation of its
double coverSpin(2d). The representation that we have constructed here on
the fermionic oscillator state spaceFd+is called the spinor representation of
Spin(2d), and we will sometimes denoteFd+with this group action asS.
In the bosonic case,H=Fdis an irreducible representation of the Heisenberg
group, but as a representation ofMp(2d,R), it has two irreducible components,
corresponding to even and odd polynomials. The fermionic analog is thatFd+
is irreducible under the action of the Clifford algebra Cliff(2d,C). One way
to show this is to show that Cliff(2d,C) is isomorphic to the matrix algebra
M(2d,C) and its action onHF=C^2
d
is isomorphic to the action of matrices
on column vectors.
WhileFd+is irreducible as a representation of the Clifford algebra, it is the
sum of two irreducible representations ofSpin(2d), the so-called “half-spinor”
representations.Spin(2d) is generated by quadratic combinations of the Clifford
algebra generators, so these will preserve the subspaces

S+= span{| 0 〉F, aF†jaF†k| 0 〉F,···}⊂S=Fd+

and
S−= span{aF†j| 0 〉F, aF†jaF†kaF†l| 0 〉F,···}⊂S=Fd+